30,224 research outputs found
Embedding nearly-spanning bounded degree trees
We derive a sufficient condition for a sparse graph G on n vertices to
contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n
vertices, in terms of the expansion properties of G. As a result we show that
for fixed d\geq 2 and 0<\epsilon<1, there exists a constant c=c(d,\epsilon)
such that a random graph G(n,c/n) contains almost surely a copy of every tree T
on (1-\epsilon)n vertices with maximum degree at most d. We also prove that if
an (n,D,\lambda)-graph G (i.e., a D-regular graph on n vertices all of whose
eigenvalues, except the first one, are at most \lambda in their absolute
values) has large enough spectral gap D/\lambda as a function of d and
\epsilon, then G has a copy of every tree T as above
Matroidal Degree-Bounded Minimum Spanning Trees
We consider the minimum spanning tree (MST) problem under the restriction
that for every vertex v, the edges of the tree that are adjacent to v satisfy a
given family of constraints. A famous example thereof is the classical
degree-constrained MST problem, where for every vertex v, a simple upper bound
on the degree is imposed. Iterative rounding/relaxation algorithms became the
tool of choice for degree-bounded network design problems. A cornerstone for
this development was the work of Singh and Lau, who showed for the
degree-bounded MST problem how to find a spanning tree violating each degree
bound by at most one unit and with cost at most the cost of an optimal solution
that respects the degree bounds.
However, current iterative rounding approaches face several limits when
dealing with more general degree constraints. In particular, when several
constraints are imposed on the edges adjacent to a vertex v, as for example
when a partition of the edges adjacent to v is given and only a fixed number of
elements can be chosen out of each set of the partition, current approaches
might violate each of the constraints by a constant, instead of violating all
constraints together by at most a constant number of edges. Furthermore, it is
also not clear how previous iterative rounding approaches can be used for
degree constraints where some edges are in a super-constant number of
constraints.
We extend iterative rounding/relaxation approaches both on a conceptual level
as well as aspects involving their analysis to address these limitations. This
leads to an efficient algorithm for the degree-constrained MST problem where
for every vertex v, the edges adjacent to v have to be independent in a given
matroid. The algorithm returns a spanning tree T of cost at most OPT, such that
for every vertex v, it suffices to remove at most 8 edges from T to satisfy the
matroidal degree constraint at v
Embedding bounded degree spanning trees in random graphs
We prove that if a tree has vertices and maximum degree at most
, then a copy of can almost surely be found in the random graph
.Comment: 14 page
Spanning Trees of Bounded Degree Graphs
We consider lower bounds on the number of spanning trees of connected graphs
with degree bounded by . The question is of interest because such bounds may
improve the analysis of the improvement produced by memorisation in the runtime
of exponential algorithms. The value of interest is the constant such
that all connected graphs with degree bounded by have at least
spanning trees where is the cyclomatic number or excess of
the graph, namely . We conjecture that is achieved by the
complete graph but we have not proved this for any greater than
3. We give weaker lower bounds on for
Proximity Drawings of High-Degree Trees
A drawing of a given (abstract) tree that is a minimum spanning tree of the
vertex set is considered aesthetically pleasing. However, such a drawing can
only exist if the tree has maximum degree at most 6. What can be said for trees
of higher degree? We approach this question by supposing that a partition or
covering of the tree by subtrees of bounded degree is given. Then we show that
if the partition or covering satisfies some natural properties, then there is a
drawing of the entire tree such that each of the given subtrees is drawn as a
minimum spanning tree of its vertex set
Minimum vertex degree conditions for loose spanning trees in 3-graphs
In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi showed that every large
-vertex graph with minimum degree at least contains all
spanning trees of bounded degree. We consider a generalization of this result
to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained
by successively appending edges sharing a single vertex with a previous edge.
We show that for all and , and large, every -vertex
3-uniform hypergraph of minimum vertex degree
contains every loose spanning tree with maximum vertex degree .
This bound is asymptotically tight, since some loose trees contain perfect
matchings.Comment: 18 pages, 1 figur
Sharp threshold for embedding combs and other spanning trees in random graphs
When , the tree consists of a path containing
vertices, each of whose vertices has a disjoint path length
beginning at it. We show that, for any and , the binomial
random graph almost surely contains
as a subgraph. This improves a recent result of Kahn,
Lubetzky and Wormald. We prove a similar statement for a more general class of
trees containing both these combs and all bounded degree spanning trees which
have at least disjoint bare paths length .
We also give an efficient method for finding large expander subgraphs in a
binomial random graph. This allows us to improve a result on almost spanning
trees by Balogh, Csaba, Pei and Samotij.Comment: 20 page
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